3 Answers
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This is a complicated question to answer in a simple yes/no matter. First there is a distinction to be made between classical and quantum mechanics. Both are deterministic theories. It is just that the determinism in quantum mechanics takes a different form. In quantum mechanics the determinism is associated with the evolution of the wave function.

It helps to understand how things evolve in classical statistical mechanics. First you have to imagine the existence of an abstract space called phase space. In phase space the instantaneous position and momentum are plotted on separate axes as shown in the hyperlink. For normal 3D space, for each particle there are 6 dimensions in phase space, 3 for position and 3 for the component of momentum associate with each position axis. If there are N particles there are 6N dimensions. At any instant, the entire state of the system can be described as a point in phase space. This is called a microstate.

Depending on the macroscopic properties of a system, there are certain microstates that are accessible by the system at any time. If the system is an isolated thermodynamic system this is called the microcanonical ensemble. If the system is in thermal equilibrium, then there is an equal probability that the system can be in any one of the accessible energy microstates. This is when the system is at maximum entropy.

If the system is classically far out of equilibrium, then there is some probability distribution function that biases the system to certain microstates. If there is a probability of one (100%) that the system has to be in one microstate out of all the microstates that have similar properties, then the system is in a low entropy condition. From a classical standpoint, the universe was in one of these low entropy conditions where only one or a few microstates was available to it at the start of the evolution.

As entropy increases, more and more microstates are available to the system. As the system evolves, from a statistical mechanics point of view, there is some probability associated with which microstate the system will find itself in next.

When the system is in thermal equilibrium, and the system is ergodic, there is a theory called Poincare Recurrence. This theory basically suggest that under certain conditions, the system can find itself back in its original starting microstate an infinite number of times. This is very controversial as it applies to the behavior of the universe, but it is likely the source of the question you have and it is true if certain assumptions are true.

This topic about the origin of the universe is unfortunately very complicated, and when we talk about quantum effects at the early stages there is a lot of theory, but limited evidence. In fact, one of the insights being provided by LHC is exactly how the universe behaved in its very early stages.

To be honest the real answer is that nobody knows for sure how the universe started, however strict adherents to quantum mechanics will tell you that classical approaches to the problem are inadequate.

As mentioned, quantum randomness drives everything. We (humans) don't see it with our eyes because the effects are small, however the effects become quite large when working with small particles.
Since the (very) early universe was nothing but a soup of insanely small unbound particles, the way in which they interacted was quite affected by the randomness.
Whether the laws of classical physics, as we know them, would be roughly the same is another story. I believe they would be much the same.
However it should be considered certain that the shape and elemental makeup of the universe would be different.